Researchers find the tipping point between resilience and collapse in complex systems

Using statistical physics, network scientist Albert-László Barabási and his colleagues have developed the first-ever tool to identify whether systems—be they technological, ecological, or biological—are in danger of failing. Photo by Matthew Moodono, Northeastern University

Hon­ey­bees have been dying in record num­bers, threat­ening the con­tinued pro­duc­tion of nutri­tious foods such as apples, nuts, blue­ber­ries, broc­coli, and onions. Without bees to pol­li­nate these crops, the envi­ron­mental ecosystem—and our health—stands in the bal­ance. Have we reached the tip­ping point, where the plant-​​pollinator system is due to collapse?

There was no way to cal­cu­late that—until now.

Using sta­tis­tical physics, North­eastern net­work sci­en­tist Albert-​​László Barabási and his col­leagues Jianxi Gao and Baruch Barzel have devel­oped a tool to iden­tify that tip­ping point—for every­thing from eco­log­ical sys­tems such as bees and plants to tech­no­log­ical sys­tems such as power grids. It opens the door to plan­ning and imple­menting pre­ven­tive mea­sures before it’s too late, as well as preparing for recovery after a disaster.

The tool, described in a new paper pub­lished on Wednesday in the pres­ti­gious journal Nature, fills a long­standing gap in sci­en­tists’ under­standing of what deter­mines “resilience”—that is, a system’s ability to adjust to dis­tur­bances, both internal and external, in order to remain functional.

“The failure of a system can lead to serious con­se­quences, whether to the envi­ron­ment, economy, human health, or tech­nology,” said Barabási, Robert Gray Dodge Pro­fessor of Net­work Sci­ence and Uni­ver­sity Dis­tin­guished Pro­fessor in the Depart­ment of Physics. “But there was no theory that con­sid­ered the com­plexity of the net­works under­lying those systems—that is, their many para­me­ters and com­po­nents. That made it very dif­fi­cult, if not impos­sible, to pre­dict the sys­tems’ resilience in the face of dis­tur­bances to those para­me­ters and components.”

“Our tool, for the first time, enables those pre­dic­tions,” said Barabási, who is also a leader in Northeastern’s Net­work Sci­ence Institute.

Taking a system’s temperature

Barzel, a post­doc­toral fellow in Barabasi’s lab who col­lab­o­rated on the research and is now at Bar-​​Ilan Uni­ver­sity, draws an ele­gant analogy between the role of tem­per­a­ture in iden­ti­fying that tip­ping point in a pot of water and the single parameter—a tem­per­a­ture equiv­a­lent, as it were—that their tool can uncover to iden­tify the tip­ping point in any com­plex system.

Baruch Barzel col­lab­o­rated on the research as a post­doc­toral fellow in Barabási’s lab. Photo by Brooks Canaday/​Northeastern University

Con­sider: 100 degrees Cel­sius is the tip­ping point for water changing from liquid to vapor. Think of liquid as the desir­able state for the system and vapor as the unde­sir­able one, sig­ni­fying col­lapse. Mil­lions of para­me­ters and com­po­nents quan­tify what is going on within that pot of water, from the rela­tion­ship of the water mol­e­cules to one another to their speed and the chem­ical bonds linking their elements.

As the water heats up, those para­me­ters and com­po­nents con­tin­u­ally change. Mea­suring those mul­ti­tudi­nous changes over time—a micro­scopic approach to assessing the water’s state—would be impos­sible. How, then, are we to know when the water is reaching the threshold that divides the desir­able (liquid) state from the unde­sir­able (vapor) state?

Simple: Using a single parameter—temperature. As the water in the pot reaches, say, 99 degrees Cel­sius, alarms go off and we know to remove it from the heat.

“Sta­tis­tical physics has found that you can crunch down all of these mil­lions of para­me­ters and com­po­nents into one number—the tem­per­a­ture,” said Barzel. “We take it for granted now, but that was a tremen­dous sci­en­tific achievement.”

The researchers’ tool sim­i­larly crunches down all the para­me­ters and com­po­nents of any com­plex system into a single cru­cial number. It enables us, essen­tially, to take the system’s “tem­per­a­ture” to deter­mine its health and respond accordingly.

Jianxi Gao, who col­lab­o­rated on the research, is a post­doc­toral fellow in Barabási’s lab. Photo by Adam Glanzman/​Northeastern University

“We col­lect all the data and map it to one number, a uni­versal resilience curve,” said Gao, a postdoc in Barabási’s lab. “That’s the only number we need in order to quan­tify whether the system is on the desir­able or unde­sir­able side of the threshold, or even approaching the danger zone.”

From theory to application

Of course, the inter­ac­tion between, say, bees and plants in an eco­log­ical system is not the same as the inter­ac­tion between water mol­e­cules in a pot, noted Barzel. “But the way of thought and the math­e­mat­ical tools that we use—statistical physics—are very similar.”

A big dif­fer­ence, how­ever, is that we know how to pre­vent the water system from col­lapsing: Turn off the heat before the tem­per­a­ture reaches 100 degrees Cel­sius. Indeed, the water system on its own pro­vides a vis­ible clue as it approaches the tip­ping point: bub­bles. You can’t say the same for the dying bees. The problem leads nat­u­rally to the researchers’ next steps: Using sta­tis­tical physics both to detect trouble in a system early on and to bring about its recovery if it has crossed the threshold.

“Once you iden­tify the rel­e­vant para­meter that con­trols the system’s resilience, you can begin to tackle how to manip­u­late that resilience—how to enhance resilience or restore resilience,” said Gao. “These are not easy ques­tions, but our theory, by giving us a pic­ture of the entire system, paves the way to the answers.”

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8 comments

These sim­pli­fi­ca­tions sound won­derful but miss the point of sta­tis­tical physics, with the use of water net­works as a model. Water evap­o­rates at ALL tem­per­a­tures and yes, it is a con­tin­uous, tem­per­a­ture depen­dent process. This process may be con­sid­ered sim­ilar to the slow col­lapse of ecosys­tems or linked net­works due to ero­sive or dis­rup­tive forces like habitat loss, long before the crit­ical point(s) of “net­work collapse”.

Con­tin­uing with the water model, we see that evap­o­ra­tion rates are sig­nif­i­cantly greater at cer­tain “tem­per­a­tures”, like boiling or melting points. How­ever, what hap­pens in a closed system as the “pres­sure” within the system increases, due to evap­o­ra­tion? It reduces the evap­o­ra­tion rate. Is there an equiv­a­lent, a “restoring” force, oper­ating in bio­log­ical and social net­works? Def­i­nitely. In biology, we rec­og­nize the resilience of the eukary­otic arche­type at the root of the resilience in Life, and its ability to rebound from major crashes or col­lapses like the great extinc­tion events.

There­fore using one number or mea­sure to define system resilience or to track impending “col­lapse” is a good pitch but sci­en­tif­i­cally ques­tion­able. Using “tem­per­a­ture”, as pro­posed in this study, or another uni­variate index– like the Shannon diver­sity index– for defining crit­ical points in the changes/​collapse of con­nected net­works is useful but lim­iting. It is an over­sim­pli­fi­ca­tion that can mis­lead both the public at large and policy makers who may not under­stand the assump­tions framing the model(s) being pre­sented to them.

For these rea­sons, I invite Barabási and his team to con­sider the envi­ron­mental and soci­etal “costs” of over­sim­pli­fi­ca­tions in sys­tems and net­work analysis, espe­cially if they are being used to drive high level policy changes.

Our work treats sys­tems that are pro­foundly dif­ferent than water in a kettle. Water mol­e­cules are sub­ject to the rules of physics — inter­acting through New­tonian (or quantum) mechanics, driven by elec­tro­mag­netic forces. The non­linear sys­tems that we treat in our work are only sim­ilar in the sense that they incor­po­rate inter­acting com­po­nents (nodes), yet these inter­ac­tions are of a very dif­ferent nature: their struc­ture is a highly dis­or­dered and the net­work is typ­i­cally het­ero­ge­neous, and their dynamics are driven by non­linear equa­tions that have no equiv­a­lent in “reg­ular” phys­ical systems.

Hence our map­ping to the water system was merely a metaphor, but was not meant to covey a com­plete math­e­mat­ical equiv­a­lence. The impor­tant aspect of this metaphor is the notion of sta­tis­tical physics that, while micro­scop­i­cally sys­tems may be extremely com­plex, their macro­scopic behavior can be cap­tured by very few large-​​scale para­me­ters, e.g., tem­per­a­ture. Indeed, the micro­scopic descrip­tion of a pot of water includes many para­me­ters, ~10^23, to be spe­cific. The loca­tion and speed of all mol­e­cules, the strength of the inter­ac­tion between each pair, and so on. Sta­tis­tical physics has taught us, how­ever, that to esti­mate how close you are to the tip­ping point of boiling, all you need to mea­sure is the pot’s tem­per­a­ture, crunching down mil­lions of para­me­ters into one number. This analogy is all we were aiming for in our ref­er­ence to boiling water — that our work has iden­ti­fied the rel­e­vant con­trol para­meter for a com­plex system’s resilience, the tem­per­a­ture ana­logue, if you want.

Is this one number the only thing that mat­ters? Clearly not. Each system has its own com­pli­ca­tions and spe­cific details that must be addressed in a con­crete fashion. But that does not mean that we cannot speak of gen­eral quan­ti­fiers of resilience. Our para­meter, \beta_​eff, has shown to be an excel­lent pre­dictor of resilience across a variety of dif­ferent sys­tems, espe­cially in deter­mining the tip­ping points of resilience loss — much like the value of tem­per­a­ture in pre­dicting the phase tran­si­tions in water. Such sim­pli­fi­ca­tions (as opposed to over­sim­pli­fi­ca­tions) work so well in sci­ence, because, indeed, for many sys­tems the macro­scopic char­ac­ter­is­tics are insen­si­tive to micro­scopic details. Phase tran­si­tions are a ter­rific example, showing many uni­versal macro­scopic obser­va­tions encom­passing sys­tems of a pro­foundly dis­tinct micro­scopic nature. Our work is no dif­ferent: it first char­ac­ter­izes the phase dia­gram for the system, and then shows that a single para­meter can be used to deter­mine what phase the system is in. These fea­tures, very large-​​scale indeed, are not sig­nif­i­cantly affected by small details, and hence they allow us to char­ac­terize, within a uni­fied frame­work, sys­tems that are oth­er­wise rather diverse.

Finally, the power of sim­pli­fi­ca­tions are not in pro­viding spe­cific, small scale appli­ca­tions, but rather in their ability to pro­vide qual­i­ta­tive, system level, insights. For instance, our paper exposes the role of struc­tural het­ero­geneity and sym­metry in enhancing resilience, large-​​scale fea­tures that have been fre­quently observed, yet their dynamic role was unclear until now. Such insights cannot be obtained by mea­suring each tree, but rather by observing the forest. Of course, exact sci­ence should not stop at that, but bal­ance between gen­er­al­i­ties and detailed val­i­da­tion, hence trans­for­ma­tive sci­ence is a com­bi­na­tion of both ends: metic­u­lous system-​​specific obser­va­tions con­verged with broad large-​​scale analysis. We believe that our work, indeed, fol­lows this com­bi­na­tion, taking a broad approach but, nonethe­less, testing its applic­a­bility down to the fine details, ana­lyzing together more than 34 net­works from three dif­ferent sci­en­tific domains, and showing con­sis­tently that our single para­meter cap­tures the cor­rect state of the system.

Won­derful! neg­a­tive tip­ping points, in ecosys­tems, seem to be reached when cer­tain key­stone species are elim­i­nated. As ecosys­tems become more sim­pli­fied through species loss, the sta­bility and resilience of the system is com­pro­mised. If this for­mula indi­cates when irre­versible failure occurs, does it not also indi­cate which species are actu­ally the key­stones of a pos­i­tive trophic cas­cade? Look what hap­pened when they rein­tro­duced wolves to Yellowstone.

Have you tried to test the for­mula using the data from the mouse and rat “utopia” experiments?

Given that tem­per­a­ture is (through Boltz­mann) a mea­sure of entropy, this sounds to me like what you are doing is in effect deter­mining the entropy of a com­plex system and then iden­ti­fying for that system that point where the total entropy of the system exceeds the energy of the system con­straints that keep it in a given phase. Is that accurate?

Would be inter­esting to apply this to eco­nomic ecosys­tems. One for example would be EURO zone. Will it col­lapse? One more spe­cific would be about Italy: is Italian eco-​​system on a verge of a crisis? ( I take Italy, but any other eco-​​system that we spot bubling could be inter­esting as well… at least from a spec­u­lator point of view…)

Sorry, I have not been able to see from the data avail­able so far a rea­son­able index (such as tem­per­a­ture for water in a closed system) that rec­on­ciles the com­peting and inter­de­pen­dent indexes involved in sys­tems that are gen­uinely com­plex. The argu­ment seems to be that such an index of indexes is the­o­ret­i­cally pos­sible. Yet that is not a method for actu­ally identifying/​creating or testing an index of indexes. The latest exam­ples in the prob­lems involved can be seen in, say, elec­tion polling in Michigan or in the deriv­a­tives mar­kets in 2007–2008.
Sim­i­larly, the con­cept of ‘col­lapse’ seems to me to be murky — there is a qual­i­ta­tive dif­fer­ence between bringing system life to an end and making system sur­vival a great deal more more dif­fi­cult, yet both are referred to as a ‘system col­lapse.’ (Con­sider ice ages, for example.) Per­haps a more accu­rate mea­sure is the ‘inte­gra­tion’ of a system and the transit point at which a system’s dis-​​integration or entropy accel­er­ates logarithmically?

Speaking as a layman, these researches are no doubt praise­worthy, and may earn degrees, grants, chairs and other perks to the researches involved – good for them.
But drawing a coarse par­allel, it com­pares to sug­gesting that the Titanic would not have sunk, if only a hypo­thet­ical sci­en­tist on board had cal­cu­lated the ten­so­rial stress applied to all the metal mol­e­cules of the ship-​​bulk.
For me and for a large con­tin­gent of cit­i­zens (regret­tably unheard, inaudible and impo­tent), we are well past the point of these aca­d­e­m­i­cally intriguing, sta­tis­tical or math­e­mat­ical niceties.
The habitat of all living things is being destroyed at an expo­nen­tially increasing rate. Every second the world has a net increase of three humans, 100 mil­lion per year. The com­pounded envi­ron­mental impact requires no com­plex cal­cu­la­tion to be deter­mined.
Hence a humble but real­istic deduc­tion springs to mind. Researches and sci­en­tists should direct their efforts to revert the path to destruc­tion. Destruc­tion is under the eyes of the sci­en­tist and of the layman. It is of little use to realize that “A great cause of the night is lack of the sun” and of little com­fort to know that what is dying will die and that when it dies is dead.

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